Free idempotent generated semigroups over bands fileFree idempotent generated semigroups over bands...

Free idempotent generated semigroups over bands

Dandan Yang

joint work with Vicky Gould

Novi Sad, June 2013

(Dandan Yang ) IG(E) over bands June 3, 2013 1 / 17

String rewriting systems

Let Σ be a finite alphabet and Σ∗ the set of all words on Σ.

A string-rewriting system R on Σ is a subset of Σ∗ × Σ∗.

A single-step reduction relation on Σ∗ is defined by

u −→ v ⇐⇒ (∃(l , r) ∈ R) (∃ x , y ∈ Σ∗) u = xly and v = xry .

The reduction relation on Σ∗ induced by R is the reflexive, transitiveclosure of −→ and is denoted by

∗−→ .

The structure S = (Σ∗,R) is called a reduction system.

∗−→ .

A word in Σ∗ is in normal form if we cannot apply a relation in R.

S is noetherian if 6 ∃ w = w0 −→ w1 −→ w2 −→ · · ·

S is confluent S is locally confluent

∗ ∗

∗ ∗ ∗ ∗

Facts:1 let ρ be the congruence generated by R. Then S is noetherian and

confluent implies every ρ-class contains a unique normal form.2 If S is noetherian, then

confluent ⇐⇒ locally confluent.

∗ ∗

∗ ∗ ∗ ∗

∗ ∗

∗ ∗ ∗ ∗

∗ ∗

∗ ∗ ∗ ∗

Facts:

1 let ρ be the congruence generated by R. Then S is noetherian andconfluent implies every ρ-class contains a unique normal form.

2 If S is noetherian, then

∗ ∗

∗ ∗ ∗ ∗

confluent implies every ρ-class contains a unique normal form.

∗ ∗

∗ ∗ ∗ ∗

confluent implies every ρ-class contains a unique normal form.

∗ ∗

∗ ∗ ∗ ∗

Free idempotent generated semigroups

Let S be a semigroup with E a set of all idempotents of S .

For any e, f ∈ E , define

e ≤R f ⇔ fe = e and e ≤L f ⇔ ef = e.

Note e ≤R f (e ≤L f ) implies both ef and fe are idempotents.

We say that (e, f ) is a basic pair if

e ≤R f , f ≤R e, e ≤L f or f ≤L e

i.e. {e, f } ∩ {ef , fe} 6= ∅; then ef , fe are said to be basic products.

Under basic products, E satisfies a number of axioms.

A biordered set is a partial algebra satisfying these axioms.

Let S be a semigroup with biordered set E .

The free idempotent generated semigroup IG(E ) is defined by

IG(E ) = 〈E : e f = ef , e, f ∈ E , {e, f } ∩ {ef , fe} 6= ∅〉.

where E = {e : e ∈ E}.

Facts:

1 φ : IG(E )→ 〈E 〉, given by eφ = e is an epimorphism.

2 E ∼= E (IG(E ))

Note IG(E ) naturally gives us a reduction system (E∗,R), where

R = {(e f , ef ) : (e, f ) is a basic pair}.

Aim Today: To study the general structure of IG(E ), for some bands.

Facts:

2 E ∼= E (IG(E ))

Facts:

2 E ∼= E (IG(E ))

Facts:

2 E ∼= E (IG(E ))

Facts:

2 E ∼= E (IG(E ))

Facts:

2 E ∼= E (IG(E ))

Facts:

2 E ∼= E (IG(E ))

IG(E ) over semilattices

IG(E ) is not necessarily regular.

Consider a semilattice

Then e f ∈ IG(E ) is not regular.

What other structures does IG(E ) might have?

IG(E ) is not necessarily regular. Consider a semilattice

Green’s star equivalences

For any a, b ∈ S ,

a L∗ b ⇐⇒ (∀x , y ∈ S1) (ax = ay ⇔ bx = by).

Dually, the relation R∗ is defined on S .

Note L ⊆ L∗, R ⊆ R∗.

Definition A semigroup S is called abundant if each L∗-class and eachR∗-class contains an idempotent of S .

a L∗ b ⇐⇒ (∀x , y ∈ S1) (ax = ay ⇔ bx = by).

Lemma (E∗,R) is locally confluent.

(i) e ≤ f , f ≤ g (ii) e ≤ f , f ≥ g

e g e f

e g e g

(iii) e ≥ f , f ≥ g (iv) e ≥ f , f ≤ g

f g e g

f g e f

Lemma (E∗,R) is locally confluent.

(i) e ≤ f , f ≤ g (ii) e ≤ f , f ≥ g

e g e f

e g e g

(iii) e ≥ f , f ≥ g (iv) e ≥ f , f ≤ g

f g e g

f g e f

Lemma (E∗,R) is noetherian.

Corollary Every element x1 · · · xn ∈ IG(E ) has a unique normal form.

Theorem IG(E ) is abundant, and so adequate.

Note Adequate semigroups belong to a quasivariety of algebras introducedin York by Fountain over 30 years ago, for which the free objects haverecently been described.

IG(E ) over simple bands

Recall that a band E is a semilattice Y of rectangular bands Eα, whereα ∈ Y .

Definition A band E is called a simple band if each Eα is either a leftzero band or a right zero band, for all α ∈ Y .Note We lose the uniqueness of normal form here! Consider the followingsimple band

a b c d e

a a b c d eb b b c e ec b b c e ed d e e d ee e e e e e

Clearly, c d = c ad = c a d = ca d = b d

Definition A band E is called a simple band if each Eα is either a leftzero band or a right zero band, for all α ∈ Y .

Note We lose the uniqueness of normal form here! Consider the followingsimple band

a b c d e

Definition A band E is called a simple band if each Eα is either a leftzero band or a right zero band, for all α ∈ Y .Note We lose the uniqueness of normal form here!

Consider the followingsimple band

a b c d e

Clearly, c d = c ad = c a d = ca d = b d(Dandan Yang ) IG(E) over bands June 3, 2013 10 / 17

Note IG(E ) does not have to be abundant.

Let E = {a, b, x , y} be a simple band with

a b x y

a a y x yb y b x yx x y x yy y y x y

Note a b does not R∗-related to any element in E .

a b x y

Let U ⊆ E (S). For any a, b ∈ S ,

a LU b ⇐⇒ (∀e ∈ U) (ae = a⇔ be = b).

a RU b ⇐⇒ (∀e ∈ U) (ea = a⇔ eb = b).

Note L ⊆ L∗ ⊆ LU , R ⊆ R∗ ⊆ RU .

Definition A semigroup S with U ⊆ E (S) is called weakly U-abundant ifeach LU -class and each RU -class contains an idempotent in U, and U iscalled the distinguished set of S .

Definition A weakly U-abundant semigroup semigroup S satisfies thecongruence condition if LU is a right congruence and RU is a leftcongruence.

a LU b ⇐⇒ (∀e ∈ U) (ae = a⇔ be = b).

a RU b ⇐⇒ (∀e ∈ U) (ea = a⇔ eb = b).

Note L ⊆ L∗ ⊆ LU , R ⊆ R∗ ⊆ RU .

a LU b ⇐⇒ (∀e ∈ U) (ae = a⇔ be = b).

a RU b ⇐⇒ (∀e ∈ U) (ea = a⇔ eb = b).

Note L ⊆ L∗ ⊆ LU , R ⊆ R∗ ⊆ RU .

a LU b ⇐⇒ (∀e ∈ U) (ae = a⇔ be = b).

a RU b ⇐⇒ (∀e ∈ U) (ea = a⇔ eb = b).

Note L ⊆ L∗ ⊆ LU , R ⊆ R∗ ⊆ RU .

Lemma For any e ∈ Eα, f ∈ Eβ, (e, f ) is basic pair in E if and only if(α, β) is a basic pair in Y .

Lemma Let θ : S −→ T be an onto homomorphism of semigroups S andT . Then a map

θ : IG(U) −→ IG(V )

defined by e θ = eθ for all e ∈ U, is a well defined homomorphism, whereU and V are the biordered sets of S and T , respectively.

Lemma For any e ∈ Eα, f ∈ Eβ, (e, f ) is basic pair in E if and only if(α, β) is a basic pair in Y .

Lemma Let θ : S −→ T be an onto homomorphism of semigroups S andT . Then a map

θ : IG(U) −→ IG(V )

defined by e θ = eθ for all e ∈ U, is a well defined homomorphism, whereU and V are the biordered sets of S and T , respectively.

Let α = w1 · · · wk with wi ∈ Eγi , for 1 ≤ i ≤ k . Then

(w1 · · · wk) θ = γ1 · · · γk .

We choose i1, j1, i2, j2, · · · , ir ∈ {1, · · · , k} in the following way:

γ1 · · · γi1−1

γi1+1 · · ·γj1 γj1+1 · · · γi2−1

γi2 · · · γir

· · · γir+1 · · · γik

We call i1, · · · , ir are the significant indices of α.

Lemma r and γi1 , · · · , γir are fixed for the equivalence class of α.

(w1 · · · wk) θ = γ1 · · · γk .

γ1 · · · γi1−1

γi1+1 · · ·γj1 γj1+1 · · · γi2−1

γi2 · · · γir

· · · γir+1 · · · γik

(w1 · · · wk) θ = γ1 · · · γk .

γ1 · · · γi1−1

γi1+1 · · ·γj1 γj1+1 · · · γi2−1

γi2 · · · γir

· · · γir+1 · · · γik

(w1 · · · wk) θ = γ1 · · · γk .

γ1 · · · γi1−1

γi1+1 · · ·γj1 γj1+1 · · · γi2−1

γi2 · · · γir

· · · γir+1 · · · γik

(w1 · · · wk) θ = γ1 · · · γk .

γ1 · · · γi1−1

γi1+1 · · ·γj1 γj1+1 · · · γi2−1

γi2 · · · γir

· · · γir+1 · · · γik

Lemma Suppose α = w1 · · · wk and β = x1 · · · xl with α ∼ β via singlereduction. Suppose that the significant indices of α and β are i1, · · · , irand z1, · · · , zr , respectively. Then w1 · · ·wi1 R x1 · · · xz1 .

Lemma Let E be a simple band and x1 · · · xn ∈ IG(E ) with normal forms

u1 · · · um = v1 · · · vs .

Then s = m and ui D vi , for all i ∈ [1,m]. In particular, we have

u1 R v1 and um L vm.

Theorem IG(E ) over a simple band is a weakly abundant semigroup withthe congruence condition.

u1 · · · um = v1 · · · vs .

Then s = m and ui D vi , for all i ∈ [1,m].

In particular, we have

u1 · · · um = v1 · · · vs .

IG(E ) over strong simple bands

Definition A band E is a strong simple band if it is a strong semilatticeof right/zero bands.

Lemma Let α = w1 · · · wk and β = x1 · · · xl such that α ∼ β via singlereduction. Suppose that the significant indices of α and β are i1, · · · , irand z1, · · · , zr , respectively. Then

x1 · · · xzl = w1 · · · wilu

where l ∈ [1, r ] and u ∈ E .

Lemma Let u1 · · · um = v1 · · · vm ∈ IG(E ) be in normal form. Then(i) ui L vi implies u1 · · · ui = v1 · · · vi ;(ii) ui R vi implies ui · · · um = vi · · · vm.

Theorem IG(E ) over a strong simple band is abundant.

Definition A band E is a strong simple band if it is a strong semilatticeof right/zero bands.Lemma Let α = w1 · · · wk and β = x1 · · · xl such that α ∼ β via singlereduction. Suppose that the significant indices of α and β are i1, · · · , irand z1, · · · , zr , respectively. Then

x1 · · · xzl = w1 · · · wilu

A key message

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